Autor

DOI:

https://doi.org/10.12775/TMNA.2021.023

Słowa kluczowe

Abstrakt

Bibliografia

H. Amann and E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (1980), 539–603.

I.V. Arnol’d, On a characteristic class entering into condition of quantization, Funct. Anal. Appl. 1 (1967), 1–13.

F. V. Atkinson, Discrete and Continuous Boundary Problems, Academic Press, London, 1964.

A. Boscaggin and M. Garrione, A note on a linear spectral theorem for a class of first order systems in R2N , Electron. J. Qual. Theory Differ. Equ. 75 (2010), 1–22.

A. Capietto and F. Dalbono, Multiplicity results for systems of asymptotically linear second order equations, Adv. Nonlinear Stud. 2 (2002), 325–356.

A. Capietto, F. Dalbono and A. Portaluri, A multiplicity result for a class of strongly indefinite asymptotically linear second order systems, Nonlinear Anal. 72 (2010), 2874–2890.

A. Capietto and W. Dambrosio, Preservation of the Maslov index along bifurcating branches of solutions of first order systems in RN , J. Differential Equations 227 (2006), 692–713.

A. Capietto, W. Dambrosio and D. Papini, Detecting multiplicity for systems of secondorder equations: an alternative approach, Adv. Differential Equations 10 (2005), 553–578.

K.C. Chang, Principal eigenvalue for weight matrix in elliptic systems, Nonlinear Anal. 46 (2001), 419–433.

P. Clément, D.G. de Figueiredo and E. Mitidieri, Positive solutions of semilinear elliptic systems, Comm. Partial Differential Equations 17 (1992), 923–940.

C. Conley, An oscillation theorem for linear systems with more than one degree of freedom, Conference on the Theory of Ordinary and Partial Differential Equations (Univ. Dundee, Dundee, 1972), Lecture Notes in Math. 280 (1972), 232–235.

C. Conley and E. Zehnder, Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math. 37 (1984), 207–253.

W.A. Coppel, Disconjugacy, Lecture Notes in Math., vol. 220, Springer–Verlag, Berlin, New York, 1971.

C. Cosner and P.W. Schaefer, Sign-definite solutions in some linear elliptic systems, Proc. Roy. Soc. Edinburgh Sect. A 111 (1989), 347–358.

D.G. Costa and C.A. Magalhães, A unified approach to a class of strongly indefinite functionals, J. Differential Equations 125 (1996), 521–547.

F. Dalbono and C. Rebelo, Multiplicity of solutions of Dirichlet problems associated with second-order equations in R2 , Proc. Edinb. Math. Soc. 52 (2009), no. 2, 569–581.

F. Dalbono and F. Zanolin, Multiplicity results for asymptotically linear equations, using the rotation number approach, Mediterr. J. Math. 4 (2007), 127–149.

E.D. da Silva, Multiplicity of solutions for gradient systems using Landesman–Lazer conditions, Abstr. Appl. Anal. 2010, Art. ID 237826, 22 pp.

D.G. de Figueiredo, Positive solutions of semilinear elliptic problems, Differential Equations (São Paulo, 1981), Lecture Notes in Mathematics, vol. 957, Springer–Verlag, 1982, pp. 34–87.

D.G. de Figueiredo and E. Mitidieri, A maximum principle for an elliptic system and applications to semilinear problems, SIAM J. Math. Anal. 17 (1986), 836–849.

D.G. de Figueiredo and P. Ubilla Superlinear systems of second-order ODE’s, Nonlinear Anal. 68 (2008), 1765–1773.

D.G. de Figueiredo and J.F. Yang, A priori bounds for positive solutions of a nonvariational elliptic system, Comm. Partial Differential Equations 26 (2001), 2305–2321.

Y. Dong, Index theory, nontrivial solutions, and asymptotically linear second-order Hamiltonian systems, J. Differential Equations 214 (2005), 233–255.

Y. Dong, Maslov type index theory for linear Hamiltonian systems with Bolza boundary value conditions and multiple solutions for nonlinear Hamiltonian systems, Pacific J. Math. 221 (2005), 253–280.

Y. Dong, Index theory for linear selfadjoint operator equations and nontrivial solutions for asymptotically linear operator equations, Calc. Var. Partial Differential Equations 38 (2010), 75–109.

I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer–Verlag, Berlin, 1990.

M.F. Furtado and F.O.V. de Paiva, Multiplicity of solutions for resonant elliptic systems, J. Math. Anal. Appl. 319 (2006), 435–449.

M.F. Furtado and F.O.V. de Paiva, Multiple solutions for resonant elliptic systems via reduction method, Bull. Aust. Math. Soc. 82 (2010), 211–220.

I.M. Gel’fand and V.B. Lidskiı̆, On the structure of the regions of stability of linear canonical systems of differential equations with periodic coefficients, Amer. Math. Soc. Transl. 8 (1958), no. 2, 143–181.

P. Gidoni and A. Margheri, Lower bounds on the number of periodic solutions for asymptotically linear planar Hamiltonian systems, Discrete Contin. Dyn. Syst. 39 (2019), 585–606.

L. Greenberg, A Prüfer Method for Calculating Eigenvalues of Self-Adjoint Systems of Ordinary Differential Equations, Part 1, Technical Report TR91-24, University of Maryland, 1991, http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.23.349.

A. Gritsans, F. Sadyrbaev and I. Yermachenko, Dirichlet boundary value problem for the second order asymptotically linear system, Int. J. Differ. Equ. (2016), Art. ID 5676217, 1–12.

A. Gritsans, F. Sadyrbaev and I. Yermachenko, Dirichlet boundary value problem for a system of n second order asymptotically asymmetric differential equations, Electron. J. Differential Equations 35 (2018), 16 pp.

Y.X. Guo, Nontrivial solutions for resonant noncooperative elliptic systems, Comm. Pure Appl. Math. 53 (2000), 1335–1349.

V.A. Jakubovic̆, Oscillatory properties of solutions of canonical equations Mat. Sb. (N.S.) 56 (1962), 3–42. (Russian)

W. Kryszewski and A. Szulkin, An infinite-dimensional Morse theory with applications, Trans. Amer. Math. Soc. 349 (1997), 3181–3234.

J. Leray and J. Schauder, Topologie et équations fonctionnelles, Ann. Sci. Ecol. Norm. Sup. 51 (1934), 45–78.

C. Li and S. Liu, Homology of saddle point reduction and applications to resonant elliptic systems, Nonlinear Anal. 81 (2013), 236–246.

V.B. Lidskiı̆, Oscillation theorems for canonical systems of differential equations, Dokl. Akad. Nauk SSSR (N.S.) 102 (1955), 877–880. (Russian)

Y. Liu, On a class of semilinear elliptic systems, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), 609–620.

C.G. Liu, Asymptotically linear Hamiltonian systems with Lagrangian boundary conditions, Pacific J. Math. 232 (2007), 233–255.

G.G. Liu, S.Y. Shi and Y.C. Wei, Multiplicity result for asymptotically linear noncooperative elliptic systems, Math. Methods Appl. Sci. 36 (2013), 1533–1542.

G.G. Liu, S.Y. Shi and Y.C. Wei, The existence of nontrivial critical point for a class of strongly indefinite asymptotically quadratic functional without compactness, Topol. Methods Nonlinear Anal. 43 (2014), 323–344.

C.G. Liu, Q. Wang and X. Lin, An index theory for symplectic paths associated with two Lagrangian subspaces with applications, Nonlinearity 24 (2011), 43–70.

C.G. Liu and Q.Y. Zhang, Nontrivial solutions for asymptotically linear Hamiltonian systems with Lagrangian boundary conditions, Acta Math. Sci. Ser. B 32 (2012), 1545–1558.

L. Lü and J. Su, Solutions to a gradient system with resonance at both zero and infinity, Nonlinear Anal. 74 (2011), 5340–5351.

A. Margheri and C. Rebelo, Multiplicity of solutions of asymptotically linear Dirichlet problems associated to second order equations in R2n+1 , Topol. Methods Nonlinear Anal. 46 (2015), 1107–1118.

A. Margheri, C. Rebelo and F. Zanolin, Maslov index, Poincaré–Birkhoff theorem and periodic solutions of asymptotically linear planar Hamiltonian systems, J. Differential Equations 183 (2002), 342–367.

M. Musso, J. Pejsachowicz and A. Portaluri, A Morse Index Theorem and bifurcation for perturbed geodesics on Semi-Riemannian Manifolds, Topol. Methods Nonlinear Anal. 25 (2005), 69–99.

A. Pomponio, Asymptotically linear cooperative elliptic system: existence and multiplicity, Nonlinear Anal. 52 (2003), 989–1003.

J. Robbin and D. Salamon, The Maslov index for paths, Topology 32 (1993), 827–844.

F. Sadyrbaev, Multiplicity of solutions for two-point boundary value problems with asymptotically asymmetric nonlinearities, Nonlinear Anal. 27 (1996), 999–1012.

Y. Shan, Multiple solutions of generalized asymptotical linear Hamiltonian systems satisfying Sturm–Liouville boundary conditions, Nonlinear Anal. 74 (2011), 4809–4819.

Y. Shan and B.Q. Liu, Multiple solutions for generalized asymptotical linear Hamiltonian systems satisfying Bolza boundary conditions, J. Appl. Math. (2013), Art. ID 521059, 5 pp.

E.A.B. Silva, Nontrivial solutions for noncooperative elliptic systems at resonance, Electron. J. Differ. Equ. Conf. 6 (2001), 267–283.

B. Sirakov, Notions of sublinearity and superlinearity for nonvariational elliptic systems, Discrete Contin. Dyn. Syst. 13 (2005), 163–174.

B. Sirakov, Existence results and a priori bounds for higher order elliptic equations and systems, J. Math. Pures Appl. (9) 89 (2008), 114–133.

M.A.S. Souto, A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems, Differential Integral Equations 8 (1995), 1245–1258.

W. Zou, Multiple solutions for asymptotically linear elliptic systems, J. Math. Anal. Appl. 255 (2001), 213–229.

Opublikowane

2022-03-06

Jak cytować

1.
Topological Methods in Nonlinear Analysis [online]. 6 marzec 2022, T. 59, nr 1, s. 163–191. [udostępniono 3.7.2024]. DOI 10.12775/TMNA.2021.023.

Numer

Dział

Articles

Statystyki

Liczba wyświetleń i pobrań: 0
Liczba cytowań: 0